---
tag: BW12J14X0Z
kind: concept
name: Manifold of smooth mappings
msc: [58D15, 46T10, 58B10]
specializes: [EBMCFQT0JN]
sources:
  - url: https://en.wikipedia.org/wiki/Banach_manifold
---

## Summary

Let $M$ be a compact manifold and $N$ a manifold.
The set $C^\infty(M, N)$ of all smooth maps is itself an infinite-dimensional manifold, modelled on [[Fréchet spaces|3GAA614673]] of smooth sections.
The charts come from the exponential map of a Riemannian metric on $N$.
A map near $f$ is written as $x \mapsto \exp_{f(x)}(s(x))$ for a small vector field $s$ along $f$.
Loop spaces, [[diffeomorphism groups|WTCNKST134]] and spaces of embeddings are open subsets or submanifolds of such manifolds.
Maps of class $C^k$, or of Sobolev class $H^s$ with $s > \dim M / 2$, form Banach or [[Hilbert manifolds|R8DMMW0E9M]] of maps, which are better suited to analysis.

## Explanations

### L1 human

A map from a circle to a sphere is a closed curve on the sphere.
All such curves together form a huge space, with one point for each curve.
Near a given curve, any nearby curve is obtained by pushing each point a little along a tangent arrow.
So nearby curves correspond to small fields of tangent arrows along the given curve.

These fields form a vector space, and this gives coordinates on the space of curves, making it a manifold of infinite dimension.